An ideal observer model of infant object perception Anonymous Author(s) Affiliation Address email Abstract Before the age of 4 months, infants make inductive inferences about the motions of physical objects. Developmental psychologists have provided verbal accounts of the knowledge that supports these inferences, but often these accounts focus on categorical rather than probabilistic principles. We propose that infant object perception is guided in part by probabilistic principles like persistence: things tend to remain the same, and when they change they do so gradually. To illustrate this idea, we develop an ideal observer model that includes probabilistic formulations of rigidity and inertia. Like previous researchers, we suggest that rigid motions are expected from an early age, but we challenge the previous claim that expectations consistent with inertia are relatively slow to develop [1]. We support these arguments by modeling four experiments from the developmental literature. Over the past few decades, ingenious experiments [1, 2] have suggested that infants rely on systematic expectations about physical objects when interpreting visual scenes. Looking time studies suggest, for example, that infants expect objects to follow continuous trajectories through time and space, and understand that two objects cannot simultaneously occupy the same location. Many of these studies have been replicated several times, but there is still no consensus about the best way to characterize the physical knowledge that gives rise to these finding. Two main approaches can be found in the literature. The verbal approach uses natural language to characterize principles of object perception [1, 3]: for example, Spelke [4] proposes that object perception is consistent with principles including continuity ("a moving object traces exactly one connected path over space and time") and cohesion ("a moving object maintains its connectedness and boundaries"). The mechanistic approach proposes that physical knowledge is better characterized by describing the mechanisms that give rise to behavior, and researchers working in this tradition often develop computational models that support their theoretical proposals [5]. We follow a third approach--the ideal observer approach [6, 7, 8]--that combines aspects of both previous traditions. Like the verbal approach, our primary goal is to characterize a set of principles that account for infant behavior, and we will not attempt to characterize the processing mechanisms that actually produce this behavior. Like the mechanistic approach, we emphasize the importance of formal models, and suggest that these models can capture forms of knowledge that are difficult for verbal accounts to handle. Most verbal theories of object perception focus on categorical principles [4], or principles that make a single distinction between possible and impossible scenes. We propose that physical knowledge in infancy also includes probabilistic principles, or expectations that make some possible scenes more surprising than others. We demonstrate the importance of probabilistic principles by focusing on two examples: the rigidity principle states that objects usually maintain their shape and size when they move, and the inertia principle states that objects tend to maintain the same pattern of motion over time. Both principles capture important regularities, but exceptions to these regularities are relatively common. 1 Focusing on rigidity and inertia will allow us to demonstrate two contributions that probabilistic approaches can make. First, probabilistic approaches can reinforce current proposals about infant perception. Spelke [3] suggests that rigidity is a core principle that guides object perception from a very early age, and we demonstrate how this idea can be captured by a model that also tolerates exceptions, such as non-rigid biological motion. Second, probabilistic approaches can identify places where existing proposals may need to be revised. Spelke [3] argues that the principle of inertia is slow to develop, but we suggest that a probabilistic version of this principle can help to account for inferences made early in development. 1 Principles of object perception Although we emphasize the importance of probabilistic principles, our model also includes five categorical principles that are closely related to principles discussed by Spelke [3, 4]. Future studies can consider whether these principles are genuinely categorical, or are better described as probabilistic expectations. For us, however, these categorical principles will provide a platform for exploring probabilistic expectations about rigidity and inertia. The categorical principles we consider rely on three basic notions: space, time, and matter. We also refer to particles, which are small pieces of matter that occupy space-time points. Particles satisfy several principles: P1. Temporal continuity. Particles are not created or destroyed. In other words, every particle that exists at time t1 must also exist at time t2 . P2. Spatial continuity. Each particle traces a continuous trajectory through space. P3. Exclusion. No two particles may occupy the same space-time point. An object is a collection of particles, and these collections satisfy two principles: P4. Discreteness. Each particle belongs to exactly one object. P5. Cohesion. At each point in time, the particles belonging to an object occupy a single connected region of space. Suppose that we are interested in a space-time window specified by a bounded region of space and a bounded interval of time. For simplicity, we will assume that space is two-dimensional, and that no particle enters or leaves the bounded region of space during the interval of interest. The collection of all particle trajectories induces a velocity field v that assigns a velocity (vx , vy ) to each point in the space-time window. Let a labeling be a function that assigns an object label to each occupied point in the space-time window, and a label of 0 to all unoccupied points. Together, a velocity field and a labeling induce a set of velocity fields {v1 , v2 , . . . , vn }, where vi is the velocity field for object i, and where n is the number of objects. We develop a theory of object perception by defining a prior distribution p(v) on velocity fields. The prior is induced by a generative process that chooses the number n of objects that will appear in the scene, then generates velocity fields {v1 , v2 , . . . , vn } for all objects in the scene. Consider first the distribution p(v1 ) on fields for a single object. Any field that violates one or more of principles P1­ P5 is assigned zero probability. For instance, fields where part of an object winks out of existence violate the principle of continuity through time, and fields where an object splits into two distinct pieces violate the principle of cohesion. Many fields, however, remain, including fields that specify non-rigid motions and jagged trajectories. For now, assume that we are working with a space of fields that is bounded but very large, and that the prior distribution over this space is uniform for all fields consistent with principles P1­P5: 0 if v1 violates P1­P5 p(v1 ) f (v1 ) = (1) 1 otherwise. Consider now the distribution p(v1 , v2 ) on fields for pairs of objects. Principles P1 through P5 rule out some of these fields, but again we must specify a prior distribution on those that remain. Our prior is induced by the following principle: P6. Independence. Velocity fields for multiple objects are independently generated subject to principles P1 through P5. 2 More formally, the independence principle specifies how the prior for the multiple object case is related to the prior p(v1 ) on velocity fields for a single object (Equation 1): 0 if {vi } collectively violate P1­P5 p(v1 , . . . , vn ) f (v1 , . . . , vn ) = (2) f (v1 ) . . . f (vn ) otherwise. 1.1 Probabilistic principles We return now to the question of prior expectations about the velocity field v1 of a single object. Principles P1­P5 make a single cut that distinguishes possible from impossible trajectories, but we need to consider whether infants have additional knowledge that makes some of the possible trajectories less surprising than others. One informal idea that seems relevant is the notion of persistence[9]: things tend to remain the same, and when they change they do so gradually. We focus on two versions of this idea that may guide expectations about velocity fields: S1. Spatial smoothness. Velocity fields tend to be smooth in space. S2. Temporal smoothness. Velocity fields tend to be smooth in time. A field is "smooth in space" if neighboring particles tend to have similar velocities at any instant in time. The smoothest possible field will be one where all particles have the same velocity at any instant--in other words, where an object is undergoing a rigid motion. The principle of spatial smoothness therefore captures the idea that objects tend to remain the same shape and size. A field is "smooth in time" if any particle tends to have very similar velocities at nearby instants of time. The smoothest possible field will be one where each particle maintains the same velocity throughout the entire interval of interest. The principle of temporal smoothness therefore captures the idea that objects tend to maintain their initial pattern of motion. For instance, stationary objects tend to remain stationary, moving objects tend to keep moving, and a moving object following a given trajectory tends to continue along that trajectory. Principles S1 and S2 are related to two principles-- rigidity and inertia--that have been discussed in the developmental literature. The rigidity principle states that objects "tend to maintain their size and shape over motion"[3], and the inertia principle states that objects move smoothly in the absence of obstacles [4]. Some authors treat these principles very differently: for instance, Spelke suggests that rigidity is one of the core principles that guides object perception from a very early age [3], but that the principle of inertia is slow to develop, and is weak or fragile once acquired. Since principles S1 and S2 seem closely related, the suggestion that one develops much later than the other seems counterintuitive. The rest of this paper explores the idea that both of these principles are needed to characterize infant perception. Our arguments will be supported by formal analyses, and we therefore need formal versions of S1 and S2. There may be different ways to formalize these principles, but we present a simple approach that builds on existing models of motion perception in adults [7, 8]. We define measures of instantaneous roughness that capture how rapidly a velocity field v varies in space and time: 2 2 O 1 v(x, y , t) v(x, y , t) Rspace (v, t) = + dxdy (3) vol(O(t)) x y (t) 1 Rtime (v, t) = vol(O(t)) O (t) v(x, y , t) t 2 dxdy (4) where O(t) is the set of all points that are occupied by the object at time t, and vol(O(t)) is the volume of the object at time t. Rspace (v, t) will be large if neighboring particles at time t tend to have different velocities, and Rtime (v, t) will be large if many particles are accelerating at time t. We combine our two roughness measures to create a single smoothness function S (·) that measures the smoothness of a velocity field: R R S (v) = -space (5) space (v, t)dt - time time (v, t)dt where space and time are positive weights that capture the importance of spatial smoothness and temporal smoothness. For all analyses in this paper we we set space = 10000 and time = 250, 3 a) L1 L2 U b) 200 0 log " p(H1 ) p(H2 ) " L1 L2 U -200 baseline smoothness Figure 1: (a) Three scenes inspired by the experiments of Spelke and colleagues [10, 11]. Each scene can be interpreted as a single object, or as a small object on top of a larger object. (b) Relative preferences for the one-object and two-object interpretations according to two models. The baseline model prefers the one-object interpretation in all three cases, but the smoothness model prefers the one-object interpretation only for scenes L1 and L2. a) H L U b) 1500 1500 1000 1000 500 500 0 0 log log log log " " p ( L) " " p ( L) p(U ) p(U ) " "p p H ( L) H ( L) H pH (U ) pH (U ) H " " X X baseline baseline smoothness smoothness Figure 2: (a) Stimuli inspired by the experiments of Spelke et al. [1]. During habituation (H), infants saw a block move behind a occluder, which was then removed to reveal the block at rest at the bottom of the display. After habituation, a barrier is added and the occluder is removed to reveal the block resting on the barrier (L) or in the same final location as before (U). A likely interpretation of each stimulus is shown: for instance, in H, the block is inferred to move smoothly from top to bottom, and in U the block shrinks to a size that allows it to navigate the gap in the barrier. (b) Relative probabilities of scenes L and U according to two models. The baseline model finds L and U equally likely a priori, but prefers U after exposure to the habituation stimulus H. The smoothness model considers L more likely even after habituation. which implies that violations of spatial smoothness are penalized more harshly than violations of temporal smoothness. We now replace Equation 1 with a prior on velocity fields that takes smooth0 ness into account: if v1 violates P1­P5 p(v1 ) f (v1 ) = (6) exp (S (v1 )) otherwise. Combining Equation 6 with Equation 2 specifies a model of object perception that can be used to make predictions about behavioral experiments. 2 Smoothness in space There are many experiments where infants aged 4 months and younger appear to make inferences that are consistent with the principle of rigidity. This section suggests that the principle of spatial smoothness can account for these results. We therefore propose that a probabilistic principle (spatial smoothness) can explain all of the findings previously presented in support of a categorical principle (rigidity), and can help in addition to explain how infants perceive non-rigid motion. One set of studies explores inferences about the number of objects in a scene. When a smaller block is resting on top of a larger block (L1 in Figure 1a), 3 month olds appear to infer that the scene includes a single object [10]. The same result holds when the small and large blocks are both moving in the same direction (L2 in Figure 1a) [11]. When these blocks are moving in opposite directions (U in Figure 1a), however, infants appear to infer that the scene contains two objects [11]. Results like these suggest that infants may have a default expectation that objects tend to move rigidly. We compared the predictions made by two models about the scenes in Figure 1a. The smoothness model uses a prior p(v1 ) that incorporates principles S1 and S2 (Equation 6), and the baseline model is identical except that it sets space = time = 0. Both models therefore incorporate principles P1­ P6, but only the smoothness model captures the principle of spatial smoothness. We have assumed so far that space and time are continuous dimensions, but now work with a discrete approximation to our model. All analyses in this paper consider scenes that unfold within a spacetime grid with 45 cells for the two spatial dimensions and 21 cells for the temporal dimension. 4 Inferring the velocity field for a scene requires computations of some sort [7, 8], but here we assume that this problem has already been solved and that the infant's perceptual system has computed a veridical velocity field v for each scene that we consider. Let H1 be the hypothesis that a given scene contains a single object, and H2 be the hypothesis that the scene contains two objects. We assume that the prior probabilities of these hypotheses are equal, and P (H1 ) = P (H2 ) = 0.5. An ideal observer can use the posterior odds ratio to choose between these hypotheses: f (v1 , v2 )dv1 dv2 P (v|H1 ) P (H1 ) f (v) P (H1 |v) = f (7) P (H2 |v) P (v|H2 ) P (H2 ) f (vA , vB ) (v1 )dv1 Equation 7 follows from Equations 2 and 6, and from approximating P (v|H2 ) by considering only the two object interpretation (vA , vB ) with maximum posterior probability. For each scene in Figure 1a, the best two object interpretation will specify a field vA for the small upper block, and a field vB for the large lower block. As described in the supplementary material, we approximate the posterior odds ratio in Equation 7 by summing over a finite space of velocity fields. Although this approximation is very rough, it allows us to explore some important qualitative differences between the smoothness and baseline models. Both models prefer the one-object hypothesis H1 when presented with scenes L1 and L2 (Figure 1b). Since there are many more two-object scenes than one-object scenes, any typical two-object interpretation is assigned lower prior probability than a typical one-object interpretation. The baseline model makes the same kind of inference about scene U, and prefers the one-object interpretation. Like infants, however, the smoothness model prefers the two-object interpretation of scene U. This model assigns low probability to a one-object interpretation where adjacent points on the object have very different velocities, and this preference for smooth motion is strong enough to overcome the simplicity preference that makes the difference when interpreting the other two scenes. Results from other experimental paradigms are also consistent with a preference for rigid motions. Figure 2 shows stimuli inspired by an experiment of Spelke et al. [1]. Four-month-old infants were habituated to a block that moved behind an occluder, and was revealed at the bottom of the display when the occluder was removed (stimulus H). After habituation, a barrier with a gap was introduced, and now the occluder was removed to reveal either the block resting on the barrier, or the block on the other side of the barrier. Looking times at these outcomes suggested that the infants were more surprised when the block appeared on the other side of the barrier, even though the block in the habituation scene always finished in that position. Scenes L and U in Figure 2 show possible explanations for the two outcomes. In likely scene L, the falling block is stopped by the barrier, but in unlikely scene U, the block shrinks to a size that allows it to pass through the barrier, then grows again until it reaches its original size. To model this experiment we need to allow for occlusion and habituation. An ideal observer can treat the occluded pixels as missing data and integrate over all possible values that they might take, and we approximate this approach by considering one or two high-probability interpretations of each occluded scene. The proper treatment of habituation is less clear, especially when the habituating stimulus also includes occlusion. We assume that an ideal observer computes the velocity field vH with maximum posterior probability given the habituating stimulus. In Figure 2, the inferred habituation field vH corresponds to a trajectory where the block falls smoothly from the top to the bottom of the scene. We now assume that the observer expects subsequent velocity fields will be similar to vH . Formally, we use a product-of-experts approach to define a post-habituation distribution on velocity fields: pH (v1 ) p(v1 )p(v1 |vH ) (8) The first expert p(v) uses the prior distribution in Equation 6, and the second expert p(v|vH ) assumes that field v is drawn from a Gaussian distribution centered on vH . More information about this model of habituation is provided in the supplementary material. The black bars in Figure 2b show predictions about L and U in the absence of any habituation. The baseline model considers both scenes to be equally probable, but the smoothness model prefers scene L since U is a case where nearby points on the falling block have different velocities. The white bars show the consequences of habituation. The smoothness model still prefers scene L, since the habituation stimulus is not enough to overwhelm its preference for rigid motion. The baseline model now prefers scene U, since the velocity field in this case is similar to the habituation field 5 a) H L U1 U2 b) log " " " " p ( L) p ( U 1) p ( L) p ( U 2) " " " " 600 400 200 0 log log log p H ( L) p H ( U 1) p H ( L) p H ( U 2) Figure 3: (a) Stimuli inspired by the experiments of [13]. The habituation stimulus H shows a block passing behind a barrier and emerging on the other side. After habituation, a new block is added either out of the direct path of the first block (L) or directly in the path of the first block (U1 and U2). In U1, the first block leaps over the second block, and in U2 the second block hops so that the first block can pass underneath. (b) Relative probabilities of scenes L, U1 and U2 according to two models. The baseline model finds all three scenes equally likely a priori, and considers L and U2 equally likely after habituation. The smoothness model considers L more likely than the other scenes both before and after habituation. a) H1 H2 L U b) " " " " " " XX X baseline smoothness log p ( L) p(U ) 300 200 log log p H 1 ( L) pH 1 (U ) p H 2 ( L) pH 2 (U ) c) 100 0 -100 baseline smoothness X X Figure 4: (a) Stimuli inspired by the experiments of Spelke et al. [14]. (b) Model predictions. After habituation to H 1, the smoothness model assigns roughly equal probabilities to L and U. After habituation to H 2, the model considers L more likely. (c) A stronger test of the inertia principle. Now the best interpretation of stimulus U involves multiple changes of direction. vH . In this and all subsequent analyses, the same habituation field vH is used for the baseline and the smoothness model, even though the smoothness model alone can infer that the habituating block moves smoothly behind the occluder. Our analysis, however, shows that the baseline model cannot account for the experimental data even when supplied with an appropriate interpretation of the habituation stimulus. Other experiments from the developmental literature have produced results consistent with the principle of spatial smoothness. For example, 4.5-month olds expect a swinging screen to be interrupted when an object is placed in its path, and 3.5-month olds are surprised when a tall object is fully hidden behind a short screen [2]. Both inferences appear to rely on the expectation that objects tend not to shrink, or to compress like foam rubber. Many of these experiments are consistent with an account that simply rules out non-rigid motion instead of introducing a graded preference for spatial smoothness. Biological motions, however, are typically non-rigid, and experiments suggest that infants can track and make inferences about objects that follow non-rigid trajectories [12]. Findings like these call for a theory like ours that incorporates a preference for rigid motion, but recognizes that non-rigid motions are possible. 3 Smoothness in time We now turn to the principle of temporal smoothness (S2) and discuss some of the experimental evidence that bears on this principle. Some researchers suggest that a closely related principle (inertia) is slow to develop, but we argue that expectations about temporal smoothness are needed to capture inferences made before the age of 4 months. Baillargeon and DeVos [13] describe one relevant experiment that explores inferences about moving objects and obstacles. During habituation, 3.5-month-old infants saw a car pass behind an occluder and emerge from the other side (stimulus H in Figure 3a). An obstacle was then placed in the direct path of the car (scenes U1 and U2) or beside this direct path (scene L), and the infants again saw the 6 car pass behind the occluder and emerge from the other side. Looking time measurements suggested that the infants were more surprised to see the car emerge when the obstacle lay within the direct path of the car. This result is consistent with the principle of temporal smoothness, which suggests that infants expected the car to maintain a straight-line trajectory, and the obstacle to remain stationary. We compared the smoothness model and the baseline model on a schematic version of this task. The black and dark gray bars in Figure 3 indicate relative a priori probabilities for scenes L, U1 and U2. The baseline model considers all three scenes equally probable, but the smoothness model prefers L. After habituation, the baseline model is still unable to account for the behavioral data, since it considers scenes L and U2 to be equally probable. The smoothness model, however, continues to prefer L. We previously mentioned three consequences of the principle of temporal smoothness: stationary objects tend to remain stationary, moving objects tend to keep moving, and moving objects tend to maintain a steady trajectory. The "car and obstacle" task addresses the first and third of these proposals, but other tasks provide support for the second. Many authors have studied settings where one moving object comes to a stop, and a second object starts to move [15]. Compared to the case where the first object collides with the second, infants appear to be surprised by the "no-contact" case where the two objects never touch. This finding is consistent with the temporal smoothness principle, which predicts that infants expect the first object to continue moving until forced to stop, and expect the second object to remain stationary until forced to start. Other experiments [16] provide support for the principle of temporal smoothness, but there are also studies that appear inconsistent with this principle. One of them uses a paradigm very similar to the "broken barrier" experiment in Figure 2 [14]. Infants are initially habituated to a block that moves from one corner of an enclosure to another (H1 in Figure 4a). After habituation, infants see a block that begins from a different corner, and now the occluder is removed to reveal the block in a location consistent with a straight-line trajectory (L) or in a location that was always the final resting place during habituation (U). Looking times suggest that infants aged 4-12 months are no more surprised by the inertia-violating outcome (U) than the inertia-consistent outcome (L). The smoothness model, however, can account for this finding. The outcome in U is contrary to temporal smoothness but consistent with habituation, and the tradeoff between these factors leads the model to assign roughly the same probability to scenes L and U (Figure 4b). Only one of the inertia experiments described by Spelke et al. [14] and Spelke et al. [1] avoids this tradeoff between habituation and smoothness. This experiment considers a case where the habituation stimulus (H2 in Figure 4a) is equally similar to the two test stimuli. The results suggest that 8 month olds are now surprised by the inertia-violating outcome, and the predictions of our model are consistent with this finding (Figure 4b). 4 and 6 month olds, however, continue to look equally at the two outcomes. Note, however, that the trajectories in Figure 4 include at most one inflection point. Experiments that consider trajectories with many inflection points can provide a more powerful way of exploring whether 4 month olds have expectations about temporal smoothness. One possible experiment is sketched in Figure 4c. The task is very similar to the task in Figure 4a, except that a barrier is added after habituation. In order for the block to end up in the same location as before, it must now follow a tortuous path around the barrier (U). Based on the principle of temporal smoothness, we predict that 4-month-olds will be more surprised to see the outcome in stimulus U than the outcome in stimulus L. This experimental design is appealing in part because previous work shows that infants are surprised by a case similar to U where the barrier extends all the way from one wall to the other [14], and our proposed experiment is a minor variant of this task. Although there is room for debate about the status of temporal smoothness, we presented two reasons to revisit the conclusion that this principle develops relatively late. First, some version of this principle seems necessary to account for experiments like the car and obstacle experiment in Figure 3. Second, most of the inertia experiments that produced null results use a habituation stimulus which may have prevented infants from revealing their default expectations, and the one experiment that escapes this objection considers a relatively minor violation of temporal smoothness. Additional experiments are needed to explore this principle, but we predict that temporal smoothness will turn out to be yet another case where knowledge is available earlier than researchers once thought. 7 4 Discussion and Conclusion We argued that characterizations of infant knowledge should include room for probabilistic expectations, and that probabilistic expectations about spatial and temporal smoothness appear to play a role in infant object perception. To support these claims, we described an ideal observer model that includes both categorical (P1 through P5) and probabilistic principles (S1 and S2), and demonstrated that the categorical principles alone are insufficient to account for several experimental findings. Our two probabilistic principles are related to principles (rigidity and inertia) that have previously been described as categorical principles. Although rigidity and inertia appear to play a role in some early inferences, formulating these principles as probabilistic expectations helps to explain how infants deal with non-rigid motion and violations of inertia. Our analysis focused on some of the many existing experiments in the developmental literature, but new experiments will be needed to explore our probabilistic approach in depth. Categorical versions of a given principle (e.g. rigidity) allow room for only two kinds of behavior, depending on whether the principle is violated or not. Probabilistic principles can be violated to a greater or lesser extent, and our approach suggests that different kinds of violations may lead to different behaviors. Future studies of rigidity and inertia can consider violations of these principles that range from mild (Figure 4a) to severe (Figure 4c), and explore whether infants respond to these violations differently. Our work builds on previous models of adult perception that rely on smoothness priors [7, 8], and the success of these models provides additional support for our approach. The results described here suggest that infant object perception is consistent with defeasible notions of rigidity and inertia, but developmental data do not provide sufficient resolution to distinguish our smoothness principles from other mathematical formulations of the same basic ideas. 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